What is the difference between a wakizashi and a katana (physically I know that the wakizashi is smaller) but what are the differences in advantages and the like?

The wakizashi is smaller. I mean, that is the difference. The wakizashi was carried by samurai as a backup in case they broke their katana.

As I recall, the wakazashi was sometimes used to commit seppuku, normally a tanto would have been used. It was carried by female members of the samurai class (they weren’t allowed to carry katanas). It may have been used in some other ceremonial functions, in place of a katana, or other blade.

Also, you could use the wakizashi indoors. Which says less about the wakizashi, and more about the techniques used to protect the katana’s (surprisingly fragile) blade. Katanas do not work well in tight spaces. Including ones that will easily accommodate other varieties of swords. So, that’s more of an inherent disadvantage for the katana, not really an advantage for the wakizashi.